An asymptotic derivation of weakly nonlinear ray theory

Using a method of expansion similar to Chapman-Enskog expansion, a new formal perturbation scheme based on high frequency approximation has been constructed. The scheme leads to an eikonal equation in which the leading order amplitude appears. The transport equation for the amplitude has been deduced with an errorO(ε²) where ε is the small parameter appearing in the high frequency approximation. On a length scale over which Choquet-Bruhat’s theory is valid, this theory reduces to the former. The theory is valid on a much larger length scale and the leading order terms give the weakly nonlinear ray theory (WNLRT) of Prasad, which has been very successful in giving physically realistic results and also in showing that the caustic of a linear theory is resolved when nonlinear effects are included. The weak shock ray theory with infinite system of compatibility conditions also follows from this theory.     

Riemann problem for kinematical conservation laws and geometrical features of nonlinear wavefronts

A pair of kinematical conservation laws (KCL) in a ray coordinatesystem (,t) are the basic equations governing the evolutionof a moving curve in two space dimensions. We first study elementarywave solutions and then the Riemann problem for KCL when themetric g, associated with the coordinate designating differentrays, is an arbitrary function of the velocity of propagationm of the moving curve. We assume that m>1 (m is appropriatelynormalized), for which the system of KCL becomes hyperbolic.We interpret the images of the elementary wave solutions inthe (,t)-plane to the (x,y)-plane as elementary shapes of themoving curve (or a nonlinear wavefront when interpreted in aphysical system) and then describe their geometrical properties.Solutions of the Riemann problem with different initial datagive the shapes of the nonlinear wavefront with different combinationsof elementary shapes. Finally, we study all possible interactionsof elementary shapes.     

高维非线性守恒律方程组

本文根据高维非线性守恒律方程组的研究历程将这一领域的研究大体分为四个阶段: 局部经典解、具扇状波结构弱解、具花状波结构弱解、整体解与混合型方程. 本文据此线索回顾与介绍多年来在该领域所获得的主要成果与进展, 并提出今后所面临的一些未解决的重要问题及困难.     

Global weakly discontinuous solutions for hyperbolic conservation laws in the presence of a boundary

This work is a continuation of our previous work, in the present paper we study the mixed initial-boundary value problem for general n×n quasilinear hyperbolic systems of conservation laws with non-linear boundary conditions in the half space . Under the assumption that each characteristic with positive velocity is linearly degenerate, we prove the existence and uniqueness of global weakly discontinuous solution u=u(t,x) with small amplitude, and this solution possesses a global structure similar to that of the self-similar solution of the corresponding Riemann problem. Some applications to quasilinear hyperbolic systems of conservation laws arising in physics and other disciplines, particularly to the system describing the motion of the relativistic string in Minkowski space R1+n, are also given.     

Lifespan of classical discontinuous solutions to general quasilinear hyperbolic systems of conservation laws with small BV initial data: Rarefaction waves

In the present paper the author investigates the global structure stability of Riemann solutions for general quasilinear hyperbolic systems of conservation laws under small BV perturbations of the initial data, where the Riemann solution contains rarefaction waves, while the perturbations are in BV but they are assumed to be C¹$C^1$-smooth, with bounded and possibly large C¹$C^1$-norms. Combining the techniques employed by Li–Kong with the modified Glimm’s functional, the author obtains a lower bound of the lifespan of the piecewise C¹$C^1$ solution to a class of generalized Riemann problems, which can be regarded as a small BV perturbation of the corresponding Riemann problem. This result is also applied to the system of traffic flow on a road network using the Aw–Rascle model.     

Limit relations for three simple hyperbolic systems of conservation laws

This paper is concerned with the limit relations from the Euler equations of one‐dimensional compressible fluid flow and the magnetohydrodynamics equations to the simplified transport equations, where the δ‐shock waves occur in their Riemann solutions of the latter two equations. The objective is to prove that the Riemann solutions of the perturbed equations coming from the one‐dimensional simplified Euler equations and the magnetohydrodynamics equations converge to the corresponding Riemann solutions of the simplified transport equations as the perturbation parameterx ε tends to zero. Furthermore, the result can also be generalized to more general situations. Copyright © 2009 John Wiley & Sons, Ltd.     

Global solutions with shock waves to the generalized Riemann problem for a system of hyperbolic conservation laws with linear damping

It is proven that a class of the generalized Riemann problem for quasilinear hyperbolic systems of conservation laws with the uniform damping term admits a unique global piecewise C¹ solution u=u(t,x) containing only n shock waves with small amplitude on t?0 and this solution possesses a global structure similar to that of the similarity solution of the corresponding homogeneous Riemann problem. As an application of our result, we prove the existence of global shock solutions, piecewise continuous and piecewise smooth solution with shock discontinuities, of the flow equations of a model class of fluids with viscosity induced by fading memory with a single jump initial data. We also give an example to show that the uniform damping mechanism is not strong enough to prevent the formation of shock waves.     

A posteriori error estimator based on derivative recovery for the discontinuous Galerkin method for nonlinear hyperbolic conservation laws on Cartesian grids

In this article, we develop and analyze a new recovery‐based a posteriori error estimator for the discontinuous Galerkin (DG) method for nonlinear hyperbolic conservation laws on Cartesian grids, when the upwind flux is used. We prove, under some suitable initial and boundary discretizations, that the ‐norm of the solution is of order , when tensor product polynomials of degree at most are used. We further propose a very simple derivative recovery formula which gives a superconvergent approximation to the directional derivative. The order of convergence is showed to be . We use our derivative recovery result to develop a robust recovery‐type a posteriori error estimator for the directional derivative approximation which is based on an enhanced recovery technique. The proposed error estimators of the recovery‐type are easy to implement, computationally simple, asymptotically exact, and are useful in adaptive computations. Finally, we show that the proposed recovery‐type a posteriori error estimates, at a fixed time, converge to the true errors in the ‐norm under mesh refinement. The order of convergence is proved to be . Our theoretical results are valid for piecewise polynomials of degree and under the condition that each component, , of the flux function possesses a uniform positive lower bound. Several numerical examples are provided to support our theoretical results and to show the effectiveness of our recovery‐based a posteriori error estimator. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1224–1265, 2017     

Global weakly discontinuous solutions to quasilinear hyperbolic systems of conservation laws with damping with a kind of non-smooth initial data

For the Cauchy problem with a kind of non-smooth initial data for weakly linearly degenerate hyperbolic systems of conservation laws with the linear damping term, we prove the existence and uniqueness of global weakly discontinuous solution u = u(t, x) containing only n weak discontinuities with small amplitude on t ≥ 0, and this solution possesses a global structure similar to that of the similarity solution of the corresponding homogeneous Riemann problem. As an application of our result, we obtain the existence and uniqueness of global weakly discontinuous solution, continuous and piecewise C ¹ solution with discontinuous first order derivatives, of the flow equations of a model class of fluids with viscosity induced by fading memory. De-Xing Kong: Supported by the National Science Foundation of China(Grant 10371073), the Special Funds for Major State Basic Research Projects of China (Grant 2000077306), the Qi Ming Xing programme of Shanghai Government, and the Project sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, the Ministry of Education of China.     

A basis of hierarchy of generalized symmetries and their conservation laws for the (3+1)-dimensional diffusion equation

We determine, by hierarchy, dependencies between higher order linear symmetries which occur when generating them using recursion operators. Thus, we deduce a formula which gives the number of independent generalized symmetries (basis) of several orders. We construct a basis for conservation laws (with respect to the group admitted by the system of differential equations) and hence generate infinitely many conservation laws in each equivalence class.     

Global structure instability of Riemann solutions for general quasilinear hyperbolic systems of conservation laws in the presence of a boundary

This work is a continuation of our previous work [Z.-Q. Shao, D.-X. Kong, Y.-C. Li, Shock reflection for general quasilinear hyperbolic systems of conservation laws, Nonlinear Anal. TMA 66 (1) (2007) 93-124]. In this paper, we study the global structure instability of the Riemann solution containing shocks, at least one rarefaction wave for general n×n quasilinear hyperbolic systems of conservation laws in the presence of a boundary. We prove the nonexistence of global piecewise C¹ solution to a class of the mixed initial-boundary value problem for general n×n quasilinear hyperbolic systems of conservation laws on the quarter plane. Our result indicates that this kind of Riemann solution mentioned above for general n×n quasilinear hyperbolic systems of conservation laws in the presence of a boundary is globally structurally unstable. Some applications to quasilinear hyperbolic systems of conservation laws arising from physics and mechanics are also given.     

Numerical resolution for a compressible two-phase flow model based on the theory of thermodynamically compatible systems

This paper is concerned with the numerical solution of the equations governing two-phase gas-solid mixture in the framework of thermodynamically compatible systems theory. The equations constitute a non-homogeneous system of nonlinear hyperbolic conservation laws. A total variation diminishing (TVD) slope limiter centre (SLIC) numerical scheme, based on the splitting approach, is presented and applied for the solution of the initial-boundary value problem for the equations. The model equations and the numerical methods are systematically assessed through a series of numerical test cases. Strong numerical evidence shows that the model and the methods are accurate, robust and conservative. The model correctly describes the formations of shocks and rarefactions in two-phase gas-solid flow.     

Existence of strong travelling wave profiles to systems of viscous conservation laws

In this paper, we prove the existence of strong travelling wave profiles for a class of viscous conservation laws when the corresponding invisid systems are hyperbolic. Besides some technical assumptions, the only main assumption is the hyperbolicity. Therefore, the existence theory can be applied to systems which are not strictly hyperbolic. Moreover, the characteristic fields can be neither genuinely nonlinear nor linearly degenerate.

     

Lie symmetry analysis,symmetry reductions with exact solutions,and conservation laws of (2+1)-dimensional Bogoyavlenskii-Schieff equation of higher order in plasma physics

In this article, similarity reductions of the (2 $+$ 1)-dimensional Bogoyavlenskii-Schieff equation of higher order have been done by Lie group method. We have determine the geometric vector field, infinitesimal generators, symmetric groups, and commutator table of Lie algebra with the help of Lie symmetry analysis. The new close form solutions and similarity solutions of the (2 $+$ 1)-dimensional Bogoyavlenskii-Schieff equation of higher order have been determined from the reduction equations. Also, the conservation laws have been derived by invoking the new conservation theorem proposed by Ibragimov.     

Research on traveling wave solutions for a class of (3+1)-dimensional nonlinear equation

Nonlinear wave phenomena are of great importance in the nature, and have became for a long time a challenging research topic for both pure and applied mathematicians. In this paper the solitary wave, kink (anti-kink) wave and periodic wave solutions for a class of (3+1)-dimensional nonlinear equation were obtained by some effective methods from the dynamical systems theory.     

On the symmetry analysis and conservation laws of the (1 + 1)‐dimensional Hénon–Lane–Emden system

We carry out a complete Lie symmetry analysis and Noether symmetry classification of the (1 + 1)‐dimensional H non–Lane–Emden system. It is shown that the principal Lie algebra, which is one dimensional, extends in several cases. It is also shown that four main cases transpire in the Noether classification with respect to the Lagrangian. In addition, conservation laws for the H non–Lane–Emden system are constructed. Furthermore, we briefly discuss the importance and the physical interpretation of these conserved vectors. Copyright © 2016 John Wiley & Sons, Ltd.     

The generalized nonlinear initial-boundary Riemann problem for linearly degenerate quasilinear hyperbolic systems of conservation laws

This work is a continuation of our previous work, in the present paper we study the generalized nonlinear initial-boundary Riemann problem with small BV data for linearly degenerate quasilinear hyperbolic systems of conservation laws with nonlinear boundary conditions in a half space . We prove the global existence and uniqueness of piecewise C¹ solution containing only contact discontinuities to a class of the generalized nonlinear initial-boundary Riemann problem, which can be regarded as a small BV perturbation of the corresponding nonlinear initial-boundary Riemann problem, for general n×n linearly degenerate quasilinear hyperbolic system of conservation laws; moreover, this solution has a global structure similar to the one of the self-similar solution to the corresponding nonlinear initial-boundary Riemann problem. Some applications to quasilinear hyperbolic systems of conservation laws arising in the string theory and high energy physics are also given.     

Soliton and Periodic Wave Solutions of the Nonlinear Loaded (3+1)-Dimensional Version of the Benjamin-Ono Equation by Functional Variable Method

In this article, we establish new travelling wave solutions for the nonlinear loaded (3+1)-dimensional version of the Benjamin-Ono equation by the functional variable method. The performance of this method is reliable and effective and the method provides the exact solitary wave solutions and periodic wave solutions. The solution procedure is very simple and the traveling wave solutions are expressed by hyperbolic functions and trigonometric functions. After visualizing the graphs of the soliton solutions and the periodic wave solutions, the use of distinct values of random parameters is demonstrated to better understand their physical features. It has been shown that the method provides a very effective and powerful mathematical tool for solving nonlinear equations in mathematical physics.